Question: The lifespans of snakes in a particular zoo are normally distributed. The average snake lives $24.8$ years; the standard deviation is $2$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a snake living less than $30.8$ years.
Solution: $24.8$ $22.8$ $26.8$ $20.8$ $28.8$ $18.8$ $30.8$ $99.7\%$ $0.15\%$ $0.15\%$ We know the lifespans are normally distributed with an average lifespan of $24.8$ years. We know the standard deviation is $2$ years, so one standard deviation below the mean is $22.8$ years and one standard deviation above the mean is $26.8$ years. Two standard deviations below the mean is $20.8$ years and two standard deviations above the mean is $28.8$ years. Three standard deviations below the mean is $18.8$ years and three standard deviations above the mean is $30.8$ years. We are interested in the probability of a snake living less than $30.8$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the snakes will have lifespans within 3 standard deviations of the average lifespan. The remaining $0.3\%$ of the snakes will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({0.15\%})$ will live less than $18.8$ years and the other half $({0.15\%})$ will live longer than $30.8$ years. The probability of a particular snake living less than $30.8$ years is ${99.7\%} + {0.15\%}$, or $99.85\%$.